A First Course in Abstract Algebra

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Considered a classic by many, A First Course in Abstract Algebra is an in-depth, introductory text which gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. The Sixth Edition continues its tradition of teaching in a classical manner, while integrating field theory and new exercises.

0 A FEW PRELIMINARIES

1

(30)

0.1 Mathematics and Proofs

1

(6)

0.2 Sets and Relations

7

(10)

0.3 Mathematical Induction

17

(4)

0.4 Complex and Matrix Algebra

21

(10)

1 GROUPS AND SUBGROUPS

31

(62)

1.1 Binary Operations

31

(12)

*Finite-State Machines (Automata)

41

(2)

1.2 Isomorphic Binary Structures

43

(8)

1.3 Groups

51

(14)

1.4 Subgroups

65

(10)

1.5 Cyclic Groups and Generators

75

(18)

Cayley Digraphs

87

(6)

2 MORE GROUPS AND COSETS

93

(68)

2.1 Groups of Permutations

93

(14)

Automata

105

(2)

2.2 Orbits, Cycles, and the Alternating Groups

107

(13)

Plane Isometries

117

(3)

2.3 Cosets and the Theorem of Lagrange

120

(8)

2.4 Direct Products and Finitely Generated Abelian Groups

128

(20)

Periodic Functions

139

(2)

Plane Isometries

141

(7)

2.5 Binary Linear Codes

148

(13)

3 HOMOMORPHISMS AND FACTOR GROUPS

161

(48)

3.1 Homomorphisms

161

(11)

3.2 Factor Groups

172

(7)

3.3 Factor-Group Computations and Simple Groups

179

(11)

3.4 Series of Groups

190

(7)

3.5 Group Action on a Set

197

(7)

3.6 Applications of G-Sets to Counting

204

(5)

4 ADVANCED GROUP THEORY

209

(44)

4.1 Isomorphism Theorems: Proof of the Jordan-Holder Theorem

209

(8)

4.2 Sylow Theorems

217

(7)

4.3 Applications of the Sylow Theory

224

(6)

4.4 Free Abelian Groups

230

(8)

4.5 Free Groups

238

(6)

4.6 Group Presentations

244

(9)

5 INTRODUCTION TO RINGS AND FIELDS

253

(72)

5.1 Rings and Fields

253

(11)

5.2 Integral Domains

264

(7)

5.3 Fermat's and Euler's Theorems

271

(6)

5.4 The Field of Quotients of an Integral Domain

277

(8)

5.5 Rings of Polynomials

285

(12)

5.6 Factorization of Polynomials over a Field

297

(11)

5.7 Noncommutative Examples

308

(8)

5.8 Ordered Rings and Fields

316

(9)

6 FACTOR RINGS AND IDEALS

325

(30)

6.1 Homomorphisms and Factor Rings

325

(9)

6.2 Prime and Maximal Ideals

334

(10)

6.3 Grobner Bases for Ideals

344

(11)

7 FACTORIZATION

355

(28)

7.1 Unique Factorization Domains

355

(13)

7.2 Euclidian Domains

368

(7)

7.3 Gaussian Integers and Norms

375

(8)

8 EXTENSION FIELDS

383

(48)

8.1 Introduction to Extension Fields

383

(10)

8.2 Vector Spaces

393

(9)

8.3 Algebraic Extensions

402

(10)

8.4 Geometric Constructions

412

(7)

8.5 Finite Fields

419

(5)

8.6 Additional Algebraic Structures

424

(7)

9 AUTOMORPHISMS AND GALOIS THEORY

431

(64)

9.1 Automorphisms of Fields

431

(10)

9.2 The Isomorphism Extension Theorem

441

(7)

9.3 Splitting Fields

448

(5)

9.4 Separable Extensions

453

(8)

9.5 Totally Inseparable Extensions

461

(4)

9.6 Galois Theory

465

(9)

9.7 Illustrations of Galois Theory

474

(7)

9.8 Cyclotomic Extensions

481

(7)

9.9 Insolvability of the Quintic

488

(7)

BIBLIOGRAPHY

495

(4)

NOTATIONS

499

(4)

ANSWERS TO ODD-NUMBERED EXERCISES NOT ASKING FOR DEFINITIONS OR PROOFS